Uniform Lyndon interpolation for the pure logic of necessitation with a modal reduction principle

We prove the uniform Lyndon interpolation property (ULIP) of some extensions of the pure logic of necessitation $\mathbf{N}$. For any $m, n \in \mathbb{N}$, $\mathbf{N}^+\mathbf{A}_{m,n}$ is the logic obtained from $\mathbf{N}$ by adding a single axiom $\Box^n φ\to \Box^m φ$, $\Diamond$-free modal reduction principle, together with a rule $\frac{\neg \Box φ}{\neg \Box \Box φ}$, required to make the logic complete with respect to its Kripke-like semantics. We first introduce a sequent calculus $\mathbf{GN}^+\mathbf{A}_{m,n}$ for $\mathbf{N}^+\mathbf{A}_{m,n}$ and show that it enjoys cut elimination, proving Craig and Lyndon interpolation properties as a consequence. We then introduce a general method, called propositionalization, that enables one to reduce ULIP of a logic to some weaker logic. Lastly, we construct a propositionalization of $\mathbf{N}^+\mathbf{A}_{m,n}$ into classical propositional logic $\mathbf{Cl}$, proving ULIP as a corollary. We also prove ULIP of $\mathbf{NA}_{m,n} = \mathbf{N} + \Box^n φ\to \Box^m φ$ and $\mathbf{NRA}_{m,n} = \mathbf{N} + \Box^n φ\to \Box^m φ+ \frac{\neg φ}{\neg \Box φ}$ in the same manner.